Complex Numbers “Not so Complex”
History • We learned in Math that we can’t have a negative number under the radical sign. • We also learned that the square of a number can’t be negative. • Now with complex numbers, these two laws no more exist.
Property of the square root of negative numbers • Lets know more about the imaginary unit ¡. 2 1 i 1. • ¡=
3 i 3
4 i 4
2i
81 1 9i
If i - 1, then i i 5
i 1 2
i i 3
i 1 4
i 1 6
i i 7
i 1 8
etc.
Examples 2
1. (i 3 ) i 2 ( 3)2 1( 3 * 3 )
1(3) 3
2. Solve 3 x 2 10 26
3 x 36 2
x 12 2
x 12 x i 12 x 2i 3 2
Complex Numbers • A complex number has a real part & an imaginary part. • Standard form is:
a bi
Real part Example: 5+4i
Imaginary part
The Complex plane
Real Axis
Imaginary Axis
Graphing in the complex plane
.
2 5i 2 2i 4 3i
4 3i
.
.
.
Adding and Subtracting (add or subtract the real parts, then add or subtract the imaginary parts) Ex: (1 2i ) (3 3i) (1 3) (2i 3i ) 2 5i
Ex: (2 3i ) (3 7i ) (2 3) (3i 7i ) 1 4i
Ex: 2i (3 i ) (2 3i ) (3 2) (2i i 3i )
1 2i
Multiplication of Complex Numbers • The product of two complex numbers is found by multiplying as if the numbers were binomials and using the fact that i2 = – 1, as follows. (a bi )(c di ) ac adi bic bidi •
ac adi bci bdi
2
ac (ad bc )i bd ( 1) (ac bd ) (ad bc )i
Multiplying Treat the i’s like variables, then change any that are not to the first power Ex: i (3 i ) 3i i 2
3i (1)
1 3i
Ex: (2 3i )(6 2i ) 12 4i 18i 6i 2 12 22i 6(1)
12 22i 6
6 22i
Conjugates • The conjugate of a – bi is a + bi The conjugate of a + bi is a – bi
3 11i 1 2i Ex : * 1 2i 1 2i (3 11i )(1 2i ) (1 2i )(1 2i ) 3 6i 11i 22i 1 2i 2i 4i 2
3 5i 22(1) 1 4(1) 3 5i 22 1 4
2
25 5i 5 25 5i 5 5
5 i
Absolute Value of a Complex Number • The distance the complex number is from the origin on the complex plane. • If you have a complex number (a bi ) the absolute value can be found using: a 2 b 2
Examples 1. 2 5i
(2) 2 (5) 2
4 25 29
2. 6i (0) 2 (6) 2
0 36
36
6 Which of these 2 complex numbers is closest to the origin?
-2+5i
Basic Concepts of Complex Numbers • Two complex numbers are equal if their real parts are equal and their imaginary parts are equal;
a bi c di if and only if
ac and bd
Basic Concepts of Complex Numbers For complex number z= a+ bi, if b = 0, then a + bi = a (real) So, the set of real numbers is a subset of complex numbers If a = 0 and b ≠0, the complex number is pure imaginary.
Developing useful rules Consider z a bi and z a bi (Conjugate) zz (a bi)(a bi) a2 b2 z
2
Remember well!!
z (a bi) (a bi) z (a bi ) (a bi) a 2 2abi b 2 a2 b2
Developing useful rules Consider z a bi and z a bi (Conjugate) z z 2a z z 2bi
Remember well!!
z 2 (a bi )(a bi ) a 2 2bi b2 z 2 (a bi )(a bi ) a 2 2abi b2
De Moivre’s Theorem ( cosq i sin q ) cos nq i sin nq n
Modulus and Argument • z= x+iy • z can be written as z= r ( cosθ + i sinθ)= reiθ
where r = |z| and θ= arg (z)
Argument • z= x+iy • z can be written as z= r ( cosθ + i sinθ) If you expand the above, you will get: Z= x+ iy = r cosθ + i r sinθ then: x= r cos θ and y= r sin θ We conclude that: cos θ = x/r and sin θ= y/r
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